Structured eigenbases and pair state transfer on threshold graphs

TL;DR

This paper characterizes threshold graphs with eigenbases of simple structure, identifies conditions for weakly Hadamard diagonalizability, and explores Laplacian pair state transfer, revealing equivalences and specific graph classes.

Contribution

It introduces a characterization of simply structured threshold graphs and their diagonalizability, and analyzes Laplacian pair state transfer on these graphs, providing new insights.

Findings

Characterization of simply structured threshold graphs.

Conditions for weakly Hadamard diagonalizability.

Equivalence of vertex and pair state transfer under certain graph joins.

Abstract

Recently, Macharete, Del-Vecchio, Teixeira and de Lima showed that a star and any threshold graph on the same number of vertices share the same eigenbasis relative to the Laplacian matrix. We use this fact to establish two main results in this paper. The first one is a characterization of threshold graphs that are \textit{simply structured}, i.e., their associated Laplacian matrices have eigenbases consisting of vectors with entries from the set $\{-1,0,1\}$. Then, we provide sufficient conditions such that a simply structured threshold graph is weakly Hadamard diagonalizable (WHD). This allows us to list all connected simply structured threshold graphs on at most 20 vertices, and identify those that are WHD. Second, we characterize Laplacian pair state transfer on threshold graphs. In particular, we show that the existence of Laplacian vertex state transfer and Laplacian pair state transfer on a threshold graph are equivalent if and only if it is not a join of a complete graph and an empty graph of certain sizes.